Notes on the Nonlinear Dependence of a Multiscale Coupled System with Respect to the Interface

NOTES ON THE NONLINEAR DEPENDENCE OF A MULTISCALE COUPLED SYSTEM WITH RESPECT TO THE INTERFACE

Fernando A Morales Escuela de Matem´aticas,Universidad Nacional de Colombia, Sede Medell´ın. Calle 59 A No 63-20, Of 43-106. Medell´ın,Colombia.

Abstract. This work studies the dependence of the solution with respect to interface geometric perturbations in a multiscaled coupled Darcy ﬂow system in direct variational formulation. A set of admissible perturbation functions and a sense of convergence are presented, as well as suﬃcient conditions on the forcing terms, in order to conclude strong convergence statements. For the rate of convergence of the solutions we start solving completely the one dimensional case using orthogonal decompositions on appropriate subspaces. Finally, the rate of convergence question is analyzed in a simple multiple dimensional setting, studying the nonlinear operators introduced by the geometric perturbations.

1. Introduction The study of saturated flow in geological porous media frequently presents natural structures with a dense network of fissures nested in the rock matrix [21]. It is also frequent to observe vuggy porous media [1], which have the presence of cavities in the rock matrix significantly larger than the average pore size of the medium. This is a multiple scale physics phenomenon, because there are regions of the medium where the flow velocity is significantly larger than the velocity on the other ones. The modeling of the interface between regions is subject to very active research: first, fluid transmission conditions across the interface are of great importance, see [19,3] for discussion of the governing laws; see [7,2] for a numerical point of view; see [1, 12,5, 16] for the analytic approach and [20,4] for a more general perspec- tive. Second, the placement of the interface is debatable since a boundary layer phenomenon between regions occurs, see [10, 18] for discussion. The interface cou- arXiv:1312.6912v3 [math.AP] 15 Mar 2014 ples regions of slow velocity (order O(1)) and fast flow (order O(1/)), see figure1. Hence, its placement and geometric description become an important issue because perturbations of the interface are inevitable. On one hand the geological strata data available are always limited, on the other hand the numerical implementation of models involving curvy interfaces, in most of the cases can only approximate the real surface. Finally, on a very different line, in the analysis of saturated flow in deformable porous media, one of the aspects is the understanding the geometric perturbations of an interface of reference.

1991 Mathematics Subject Classiﬁcation. Primary: 58F15, 58F17; Secondary: 53C35. Key words and phrases. Multiscale coupled systems, interface geometric perturbations, variational formulations, nonlinear dependence. The author was supported by the project HERMES 17194 from Universidad Nacional de Colombia. 1 2 FERNANDO A MORALES

Figure 1. Original Domain and a Perturbation

Ωζ 2 Ω = k 2 2 k ϵ = k 2 ζ k ϵ Γ

Γ Γ = k k 1 k=k Ωζ 1 Ω 1 1

Clearly, the continuity of the solution with respect to the geometry of the interface is an important issue which has received very little treatment and mainly limited to flat interfaces. Most of the theoretical achievements in the field of multiscale coupled systems, concentrate their efforts in removing the singularities introduced by the scales using homogenization processes. These techniques can be either formal [17, 13], analytic [11] or numerical [14]. Here, we model the stationary problem with a coupled system of partial differen- tial equations of Darcy flow in both regions, in direct variational formulation. We 1 simulate the region of fast flow scaling by the ratio of permeability over viscos- ity, as in figure1. It will be assumed that the real interface Γ is horizontal flat and the perturbed one Γζ is curved. Of course a flat surface will not be perturbed when discretized and seems unrealistic to consider perturbations of it. Our choice is motivated by two reasons: first, for the sake of clarity in the notation, calculation and interpretation of the results. Second, when studying the phenomenon of saturated flow in deformable porous media perturbations of a flat surface are of interest. Finally it is important to highlight that the mathematical essentials of the problem are captured in this framework. The paper starts proving in section 2, that the solutions depend continuously with respect to the interface, then it moves to the much deeper question of exploring the dependence itself. In section 3 the one dimensional case the rate of convergence question is solved completely using orthogonal decomposition of adequate subspaces, finally section 4 reveals the highly nonlinear dependence of the solutions with respect to the interface. We close this section introducing the notation. Vectors are denoted by boldface letters, as are vector-valued functions and corresponding function spaces. We use xe to indicate a vector in RN−1; if x ∈ RN then the RN−1 ×{0} projection is identified def with xe = (x1, x2, . . . , xN−1) so that x = (xe, xN ). The symbol ∇e represents the RN R R gradient in the first N − 1 derivatives. Given a function f : → then M f dS RN−1 RN R is the notation for its surface integral on the manifold M ⊆ . A f dx stands for the volume integral in the set A ⊆ RN ; whenever the context is clear INTERFACE PERTURBATIONS AND NON-LINEARITY 3

R 2 we simply write A f. The notation k · k0,A, k · k1,A respectively denote the L (A) 1 N and H (A) norms on the domain A ⊆ R . 1A stands for the indicator function of any given set A. The be` indicates the unitary vector in the `-th direction for RN 1 ≤ ` ≤ N. We denote by νb the outwards normal vector to smooth domain in and n indicates the upwards normal vector i.e. n · eN ≥ 0. Finally, the Lebesgue b N b b measure in R is denoted by λN .

2. Formulation and Convergence 2.1. Geometric Setting. In the following Γ denotes a connected set in RN−1×{0} whose projection onto RN−1 is open; from now on we make no distinction between these two domains. Similarly, Ω 1, Ω 2 denote be smooth bounded open regions in RN i separated by Γ, i.e. ∂Ω 1 ∩ ∂Ω 2 = Γ; and such that sgn(x · ebN ) = (−1) for each x ∈ Ωi, i = 1, 2 (see ﬁgure1). Next, we introduce the admissible perturbations of the interface Γ. Deﬁnition 2.1. We say the set T (Γ, Ω) of piecewise C1 perturbations of the interface Γ contained in Ω is given by

def (1) T (Γ, Ω) = {ζ ∈ C Γ :(xe, ζ(xe )) ∈ Ω ∀ xe ∈ Γ 1 , ζ |∂Γ = 0 and ζ is a piecesise C function }. The interface associated to ζ ∈ T (Γ, Ω) is given by the set ζ def RN (2a) Γ = {(xe, ζ(xe )) ∈ : xe ∈ Γ}. The domains associated to ζ ∈ T (Γ, Ω) are deﬁned by the sets ζ def (2b) Ω 1 = {(xe, xN ) ∈ Ω: xe ∈ Γ, xN < ζ(xe )}

ζ def (2c) Ω 2 = {(xe, xN ) ∈ Ω: xe ∈ Γ, ζ(xe ) < xN } Remark 1. Observe the following facts ζ ζ ζ (3a) ∂Ω 1 ∩ ∂Ω 2 = Γ ,

ζ ζ ζ (3b) Ω 1 ∪ Γ ∪ Ω 2 = Ω,

ζ ζ (3c) ∂Ω 1 − Γ = ∂Ω 1 − Γ,

ζ ζ (3d) ∂Ω 2 − Γ = ∂Ω 2 − Γ. Deﬁnition 2.2. Deﬁne the space

def 1 (4) V = {u ∈ H (Ω) : u| ∂Ω1−Γ = 0} Endowed with the inner product h·, ·i : V × V → R

def Z (5) hu, viV = ∇u · ∇v, Ω def p and the norm kukV = hu, uiV . Remark 2. Recall that due to the boundary condition deﬁning the space V and 1 the Poincar´einequality the k · kV -norm is equivalent to the standard H -norm. 4 FERNANDO A MORALES

2.2. The Problems. Consider the strong problem k (6a) − ∇ · i ∇p = F in Ω , i = 1, 2, i−1 i i with the interface conditions k (6b) p = p , k ∇p · n − 2 ∇p · n = f on Γ, 1 2 1 1 b 2 b and the boundary conditions

(6c) p1 = 0 on ∂Ω1 − Γ, ∇p2 · nb = 0 on ∂Ω2 − Γ. Now, its perturbation in strong form is given by k (7a) − ∇ · i ∇q = F in Ωζ , i = 1, 2, i−1 i i with the interface conditions k (7b) q = q , k ∇q · n − 2 ∇q · n = f on Γ ζ , 1 2 1 1 b 2 b and the boundary conditions ζ ζ ζ ζ (7c) q1 = 0 on ∂Ω1 − Γ , ∇q2 · nb = 0 on ∂Ω2 − Γ . Both systems above model stationary Darcy flow, coupling the regions depicted in the left and right hand side of figure1 respectively. The coefficients k1, k2 indicate the permeability in the corresponding domain; for simplicity they will be omitted 1 1 in the following. The scaling factor ensures a much higher velocity O( ) in the upper region with respect to the lower region fluid velocity O(1). The term F stands for fluid sources and f for a normal flux forcing term on the interface; it is assumed that f is well defined L2-function on both manifolds Γ and Γζ . Hence, the weak problems in direct formulation are given by Z 1 Z Z Z (8a) p ∈ V : ∇p · ∇r + ∇p · ∇r = F r + f r d S , ∀ r ∈ V Ω1 Ω2 Ω Γ Z 1 Z Z Z (8b) q ζ ∈ V : ∇q ζ · ∇r + ∇q ζ · ∇r = F r + f r d S , ∀ r ∈ V ζ ζ ζ Ω1 Ω2 Ω Γ Theorem 2.3. The problems (8a), (8b) are well-posed. Proof. Is is a direct application of Lax-Milgram’s lemma and the Poincaréinequal- ity, see [15] for details. 2.3. A-priori Estimates and Weak Convergence. In this section under rea- sonable conditions on the forcing terms and the appropriate type of convergence for the perturbations ζ, a-priori estimates on the solutions of problems (8b) as well as weak convergence statements to the solution of problem (8a) are attained. Test equation (8b) with the solution qζ , due to the boundary conditions of V and the

Poincar´econstant CΩ we get

1 ζ 2 ζ 2 ζ 2 1 ζ 2 (9) k q k1, Ω ≤ k∇q k0,Ω ≤ k∇q k ζ + k∇q k ζ 2 0,Ω1 0,Ω2 1 + CΩ Z ζ ζ ≤ kF k0,Ω kq k0,Ω + f q dS. Γζ INTERFACE PERTURBATIONS AND NON-LINEARITY 5

If a sequence of perturbations {ζn} ⊆ T (Γ, Ω) is to be analyzed, conditions on the type of convergence must be speciﬁed. For the perturbations, we assume that n o N (10a) ess sup |(−∇e ζn(xe), 1)| : xe ∈ Γ ≤ C0 ∀ n ∈ . i.e. the gradients are globally bounded. Additionally assume uniform convergence

(10b) kζnkC(Γ) −−−−→ 0 n→∞ From now on we denote Γn = Γζn and qn = qζn . For the forcing terms we assume there exists an open set G containing {Γn} and an element Φ ∈ H div(G) with G an open region such that Φ · nb|Γn = f for all n. n Here nb denotes the upwards normal vector to Γ and 2 2 H div = {v ∈ L (G): ∇ · v ∈ L (G)}. Deﬁne n def [ (11) U = (0, ζn(xe)) ∪ (ζn(xe), 0) xe∈Γ then, ∂U n = Γn ∪Γ. Due to condition (10a) the domain U n has Lipschitz boundary, then the classical duality relationship [22] holds i.e. Z Z Z Z n n n n n f q dS − f q dS = Φ · νb q dS = ∇ · Φ q + Φ · ∇q Γn Γ ∂U n U n then Z n n n (12) f q dS ≤ kfk 1 kq k 1 + kΦkH (Γ)kq k1,Ω − 2 ,Γ 2 ,Γ div Γn n ≤ {kfk 1 + kΦkH (Γ)}kq k1,Ω. − 2 ,Γ div Combining (12) with (9) gives n (13) k q k1, Ω ≤ {CΩkF k0,Ω + kfk 1 + kΦkH (Γ)}. − 2 ,Γ div Due to the Rellich-Kondrachov theorem, there must exist a subsequence, denoted {qk} and an element q∗ ∈ H1(Ω) such that qk → q∗ weekly in H1(Ω), strongly in L2(Ω).

k ζk Denoting Ωi = Ωi for i = 1, 2, the variational statement can be written as Z Z Z Z k 1 k (14) ∇q · ∇r 1 k + ∇q · ∇r 1 k = F r + f r d S Ω1 Ω2 Ω Ω Ω Γk 1 for r ∈ V arbitrary. Lett ζk → 0 in C(Γ), ﬁrst observe that for any r ∈ H (Ω) holds Z Z Z

f r dS − f r dS = ∇ · Φ r + Φ · ∇r ≤ kΦk k krk 1 k H div(U ) H (U ) Γk Γ U k R Since the right hand side converges to 0 as k → ∞ it follows that Γk f r dS → R 2 f r dS. Next observe that {∇r 1 k } converges strongly in L (Ω); together with Γ Ωi the convergence of the surface forcing terms previously discussed, the expression (14) converges to Z Z Z Z ∗ 1 1 ∗ 1 ∇q · ∇r Ω1 + ∇q · ∇r Ω2 = F r + f r dS. Ω Ω Ω Γ Since q∗ is in V and the variational statement above holds for all r ∈ V the unique- ness of problem (8a) implies q∗ = p. The reasoning above holds for any subsequence 6 FERNANDO A MORALES of {qn} and the solution of (8a) is unique, then it follows that the whole sequence converges to p i.e. (15) qn → p weekly in H1(Ω), strongly in L2(Ω). We close the section with an important observation. Test the statements (8b) on the diagonal qn and let n → ∞; it yields ( ) Z 1 Z Z Z (16) lim |∇qn|2 + |∇qn|2 = F p + f p dS n→∞ n n Ω1 Ω2 Ω Γ Z 1 Z = |∇p|2 + |∇p|2. Ω1 Ω2

R 2 1 R 2 1 The map r 7→ { |∇r| + |∇r| } 2 is a norm equivalent to the norm k · kV . Ω1 Ω2 n However, due to the presence of the domains Ωi , i = 1, 2 the equality (16) is not a statement of norms convergence which, together with the weak convergence, would allow to conclude strong convergence. However, due to the weak convergence and R 2 1 R 2 1 the equivalence of the norms r 7→ { |∇r| + |∇r| } 2 and k · kV we can Ω1 Ω2 conclude that Z 1 Z Z 1 Z (17) |∇p|2 + |∇p|2 ≤ lim inf |∇qn|2 + |∇qn|2 . n Ω1 Ω2 Ω1 Ω2

2.4. The Strong Convergence. Given a function r ∈ V consider the following identities Z Z Z Z (18a) |∇r|2 = |∇r|2 − |∇r|2 + |∇r|2 ζ ζ ζ Ω1 Ω1 −Ω1 Ω2 −Ω2 Ω1

Z Z Z Z (18b) |∇r|2 = |∇r|2 − |∇r|2 + |∇r|2 ζ ζ ζ Ω2 Ω2 −Ω2 Ω1 −Ω1 Ω2 We deﬁne the perturbation term as Z Z def 2 2 (19) Ξζ (r) = |∇r| − |∇r| ζ ζ Ω2 −Ω2 Ω1 −Ω1 Moreover, the perturbation term satisﬁes that

(20) Z Z Z Z 2 2 2 2 |Ξζ (r)| = |∇r| − |∇r| ≤ |∇r| + |∇r| ζ ζ ζ ζ Ω2 −Ω2 Ω1 −Ω1 Ω2 −Ω2 Ω1 −Ω1 2 h ζ ζ i ≤ krkV λN (Ω1 − Ω1) + λN (Ω2 − Ω2) . Therefore the following estimate holds Z Z Z Z 2 1 2 2 1 2 1 |∇r| + |∇r| = |∇r| + |∇r| + 1 − Ξζ (r) ζ ζ Ω1 Ω2 Ω1 Ω1 Z Z 2 1 2 1 h ζ ζ i 2 ≥ |∇r| + |∇r| − 1 − λN (Ω1 − Ω1) + λN (Ω2 − Ω2) krkV . Ω1 Ω1 INTERFACE PERTURBATIONS AND NON-LINEARITY 7

We know krk2 ≥ {R |∇r|2 + 1 R |∇r|2} then, combining with the expression V Ω1 Ω2 above we get Z 1 Z |∇r|2 + |∇r|2 ζ ζ Ω1 Ω2 Z Z 1 h ζ ζ i 2 1 2 ≥ 1 − 1 − λN (Ω1 − Ω1) + λN (Ω2 − Ω2) |∇r| + |∇r| . Ω1 Ω1 Deﬁning

def 1 h ζ ζ i (21) Cζ = 1 − 1 − λN (Ω − Ω1) + λN (Ω − Ω2) , 1 2 We get the estimate Z Z Z Z 2 1 2 2 1 2 (22) |∇r| + |∇r| ≥ (1 − Cζ ) |∇r| + |∇r| ∀ r ∈ V. ζ ζ Ω1 Ω2 Ω1 Ω1 In particular for the sequence of solutions {qn : n ∈ N} ⊆ V holds Z Z Z Z n 2 1 n 2 n 2 1 n 2 Cζn |∇q | + |∇q | ≤ |∇q | + |∇q | n n Ω1 Ω2 Ω1 Ω2

Letting n → ∞ gives kζnkC(Γ) → 0 and consequently Cζn → 0. Then, taking lim supn in the expression above yields Z 1 Z (23) lim sup |∇qn|2 + |∇qn|2 n Ω1 Ω2 ( ) Z 1 Z Z 1 Z ≤ lim sup |∇qn|2 + |∇qn|2 = |∇p|2 + |∇p|2. n n n Ω1 Ω2 Ω1 Ω2 Where the last equality holds due to (16). Putting together (17) and (23) we conclude Z 1 Z Z 1 Z (24) lim |∇qn|2 + |∇qn|2 = |∇p|2 + |∇p|2. n Ω1 Ω2 Ω1 Ω2 R 2 1 R 2 1 i.e. the norms r 7→ { |∇r| + |∇r| } 2 converge and, due to the equivalence Ω1 Ω2 n n with the V -norm it follows that kq kV | → kpkV . Since q converges weakly to p in V it follows that n 2 (25) kq − pkV → 0 as n → ∞.

3. The One Dimensional Case Here, we restrict our attention to the one dimensional problem in order to gain deep insight on the phenomenon. An example of how valuable this approach is, can def be found in [6]. For the problem in one dimensional setting we choose Ω 1 = (−1, 0), def Ω 2 = (0, 1) and the interface Γ = {0}. In this context a perturbation is given by ζ def ζ def a single point ζ; the perturbed domains are given by Ω 1 = (−1, ζ), Ω 2 = (ζ, 1) ζ def ζ ζ ζ and the perturbed interface Γ = {ζ}. Clearly Ω = Ω1 ∪ Γ ∪ Ω2 = Ω 1 ∪ Γ ∪ Ω 2 = (−1, 1), see ﬁgure2. Notice that in the one dimensional case the space V and its inner product given in deﬁnition 2.2 reduce to (26a) V = r ∈ H1(−1, 1) : r(−1) = 0 , 8 FERNANDO A MORALES

Z 1

(26b) hπ, κiV = ∂π ∂κ. −1 Where ∂ indicates de weak derivative. Similarly, the problems (8b) and (8b) trans- form in Z 0 1 Z 1 Z 1 (27a) p ∈ V : ∂p ∂r + ∂p ∂r = F r + f (0) r (0) ∀ r ∈ V. −1 0 −1

Z ζ 1 Z 1 Z 1 (27b) q ∈ V : ∂q ∂r + ∂q ∂r = F r + f (ζ) r (ζ) ∀ r ∈ V. −1 ζ −1 3.1. The Subspace H and its Orthogonal Projection. In order to estimate ζ ζ ζ the norms kp − q k 1,Ω, kp − q k 0,Ω we need to project the solutions p and q into the adequate subspace using the convenient geometry deﬁned by the inner product (26b). For simplicity, from now on it will be assumed ζ > 0. Consider the subspaces

(28a) H def= {κ ∈ V : ∂κ = 0 on (0, ζ)}

⊥ def (28b) H = {π ∈ V : hπ, κiV = 0 ∀ κ ∈ H} Next, we characterize the structure of H⊥. Lemma 3.1. Let H⊥ and H deﬁned in (28) then (29) H⊥ = {π ∈ V : π = 0 on (−1, 0) , ∂π = 0 on (ζ, 1)} Proof. It is direct to see that if π ∈ V is such that π = 0 in (−1, 0) and ∂π = 0 ⊥ ∞ in (ζ, 1) then π ∈ H . For the other inclusion take ρ ∈ C 0 (−1, 0) such that R 0 −1 ρ dx = 1 and extended it by zero to the whole domain (−1, 1). Choose any ∞ φ ∈ C 0 (−1, 0), extended it by zero to (−1, 1) and build the auxiliary function x 0 x def Z Z Z Φ(x) = φ(t) d t 1(−1,0) (x) − φ(y) dy ρ(t) d t 1(−1,0) (x). −1 −1 −1 It is direct to see that Φ ∈ H. Now take any π ∈ H⊥, then Z 0 Z 0 Z 0

0 = hπ, ΦiV = ∂π ∂Φ = ∂π (x) φ (x) − φ(y) dy ρ(x) dx −1 −1 −1 Z 0 Z 0 Z 0 = ∂π (x) φ (x) dx − φ(y) dy ∂π (x) ρ(x) dx. −1 −1 −1 i.e. Z 0 Z 0 Z 0 ∂π (x) φ (x) dx = φ(y) dy ∂π (x) ρ(x) dx, −1 −1 −1 ∞ for all φ ∈ C 0 (−1, 0). Therefore, we conclude ∂π must be constant in (−1, 0). Using an analogous construction we also conclude ∂π must be constant in (ζ, 1). Now we prove that such constants must be zero. Consider any κ ∈ H, then we have Z 0 Z 1 Z 0 Z 1

0 = hπ, κiV = ∂π ∂κ + ∂π ∂κ = ∂π(−1) ∂κ + ∂π(1) ∂κ −1 ζ −1 ζ = ∂π(−1) (κ (0) − κ (−1)) + ∂π(1) (κ (1) − κ (ζ)) . Since κ ∈ H ⊂ V it holds κ (−1) = 0 and the above expression writes (30) ∂ π(−1) κ (0) + ∂ π(1) (κ (1) − κ (ζ)) = 0. INTERFACE PERTURBATIONS AND NON-LINEARITY 9

Due to ∂κ = 0 on (0, ζ) the function κ must be constant on this interval, therefore κ (0) = κ (ζ). Recalling the above holds for any κ ∈ H, choose a test function such that κ (0) = κ (ζ) = 0 and κ(1) 6= 0, then (30) reduces to ∂π(1) κ (1) = 0 and we conclude ∂π(1) = 0. Hence, (30) reduces to ∂π(−1) κ(0) = 0. Since κ ∈ H is arbitrary we know κ(0) need not be zero for all κ ∈ H, then we conclude ∂π(−1) = 0. Therefore π must be constant on the intervals (−1, 0) and (ζ, 1). Finally, the fact that π ∈ H ⊥ ⊂ V yields π(−1) = 0; this implies π = 0 on (−1, 0) which completes the proof. Now we present the characterization of the orthogonal projections onto the subspaces H and H⊥.

⊥ Theorem 3.2. Let H,H be the spaces deﬁned in (28). Denote PH and PH⊥ the orthogonal projections onto the subspaces H and H⊥ respectively. Then, for any r ∈ V holds 1 1 1 (31a) PH r(x) = r(x) [−1, 0 ](x) + r (0) [ 0, ζ ](x) + {r(x) − [r (ζ) − r (0)]} [ζ, 1](x)

1 1 (31b) PH⊥ r(x) = [r(x) − r (0)] [ 0, ζ ](x) + [r (ζ) − r (0)] [ζ, 1](x)

Where 1A(·) denotes the indicator function of the set A.

Proof. For any r ∈ V it is direct to see that the function x 7→ r(x)1[−1, 0 ](x) + r (0) 1[ 0, ζ ](x) + {r(x) − [r (ζ) − r (0)]} 1[ζ, 1](x) is in H and that the map x 7→ ⊥ [r(x) − r (0)] 1[ 0, ζ ](x) + [r (ζ) − r (0)] 1[ζ, 1](x) belongs to H . Also, their sum gives r. The result follows due to the characterization given in lemma 3.1. Remark 3. In order to better understand the nature of the orthogonal decomposition we present ﬁgure2. An absolutely continuous function r (blue line) is decomposed in PH r (turquoise line) and PH⊥ r = (I − PH )r (red line). Also, the ζ ζ domains Ω1, Ω2 and Ω1, Ω2 are depicted.

3.2. The Problems Restricted to H. Test the problem (27a) with a function κ ∈ H, we have Z 0 1 Z 1 Z 1 ∂p ∂κ + ∂p ∂κ = F κ + f (0) κ (0) . −1 ζ −1

Now decompose p in PH p and PH p using (31a) and (31b); we get " # Z 0 1 Z 1 Z 0 Z 1 Z ζ ∂ (PH p) ∂κ + ∂ (PH p) ∂κ = F κ + F κ + F + f (0) κ (0) . −1 ζ −1 ζ 0 Here, the last equality used the fact that κ is constant in (0, ζ) for all κ ∈ H. We write the statement as

Z 0 1 Z 1 (32) PH p ∈ H : ∂ (PH p) ∂κ + ∂ (PH p) ∂κ −1 ζ Z 0 Z 1 "Z ζ # = F κ + F κ + F + f (0) κ (0) , ∀ κ ∈ H. −1 ζ 0 10 FERNANDO A MORALES

Figure 2. Orthogonal Decomposition

−1 r ζ 1 0 Ω Ω 1 2

ζ −ζ 1 1

P H r

−1 0 ζ 1

−1 0 ζ 1 ( − ) I P H r

Ωζ Ωζ 1 2

On the other hand consider the problem

Z 0 1 Z 1 (33) σ ∈ H : ∂σ ∂κ + ∂σ ∂κ −1 ζ Z 0 Z 1 "Z ζ # = F κ + F κ + F + f (0) κ (0) , ∀ κ ∈ H. −1 ζ 0

def R 0 1 R 1 The bilinear the form A (π, κ) = −1 ∂ π ∂κ + ζ ∂π ∂κ is H-elliptic and continuous, therefore the problem (33) is well-posed and we conclude that PH p is the unique solution of (33). Repeating the same procedure on the perturbed problem

(27b) we conclude PH q is the unique solution to the well-posed variational problem

Z 0 1 Z 1 (34) PH q ∈ H : ∂ (PH q) ∂κ + ∂ (PH q) ∂κ −1 ζ Z 0 Z 1 "Z ζ # = F κ dx + F κ + F + f (ζ) κ (ζ) , ∀ κ ∈ H. −1 ζ 0

3.3. The Problems Restricted to H⊥. We repeat the same strategy of the previous section and get

Z ζ Z ζ Z 1 ⊥ 1 ⊥ (35) PH⊥ p ∈ H : ∂ (PH⊥ p) ∂κ = F κ + F κ (ζ) , ∀ κ ∈ H . 0 0 ζ INTERFACE PERTURBATIONS AND NON-LINEARITY 11

This is the weak solution of following strong problem 1 −∂ ∂P ⊥ p = F in (0, ζ) , H

PH⊥ p = 0 in [−1, 0) , (36) PH⊥ p = constant in (ζ, 1) , Z 1 1 − ∂PH⊥ p ζ = F. ζ − Where ∂PH⊥ p (ζ ) = lim t → ζ− ∂PH⊥ p (t). In the same fashion Z ζ Z ζ Z 1 ⊥ ⊥ (37) PH⊥ q ∈ H : ∂ (PH⊥ q) ∂κ = F κ + F + f (ζ) κ(ζ), ∀ κ ∈ H . 0 0 ζ Where the simpliﬁcation on the term of the right hand side has been made since

κ (x) = κ (ζ) for x ∈ (ζ, 1). Thus PH⊥ q is the solution to the strong problem

−∂ ∂PH⊥ q = F in (0, ζ) ,

PH⊥ q = 0 in [−1, 0) , (38) PH⊥ q = constant in (ζ, 1) , Z 1 − ∂PH⊥ q ζ = F + f (ζ) . ζ

3.4. Estimates for the H Projections. Test (32) and (34) with PH p − PH q and subtract the result to get

Z 0 1 Z 1 ∂(PH p − PH q) ∂(PH p − PH q) + ∂(PH p − PH q) ∂(PH p − PH q) −1 ζ

= f (0) [PH p(0) − PH q(0)] − f (ζ)[PH p(ζ) − PH q(ζ)]

= [f (0) − f (ζ)] [PH p(0) − PH q(0)] .

√The last equality holds true since κ (0) = κ (ζ) for all κ ∈ H. Since |r(x)| ≤ 2 k∂rk0,(−1,1) for all r ∈ V . Thus, the expression above can be estimated by

(39) k PH p − PH qkV ≤ C |f (0) − f(ζ)| 3.5. Estimates for the H⊥ Projections. Since (36) and (38) are both ordinary diﬀerential equations, the exact solutions can be found. These are given by Z xZ t 1 (40) PH⊥ p(x) = − F (s) ds dt [0, ζ ](x) 0 0 Z 1 "Z ζZ t Z 1 # + x F 1[0, ζ ](x) − ζ F (s) ds dt − F 1[ζ, 1](x) 0 0 0 0

Z xZ t Z 1 1 1 (41) PH⊥ q(x) = − F (s) ds dt [0, ζ ](x) + F + f(ζ) x [0, ζ ](x) 0 0 0 (Z ζZ t Z 1 ) − F (s) ds dt − F + f(ζ) ζ 1[ζ, 1](x) 0 0 0 12 FERNANDO A MORALES

We estimate the norm of the diﬀerence using the exact expressions (40) and (41). When computing the L 2-norm of the diﬀerence of derivatives we get

Z (·) kP ⊥ p − P ⊥ qk = k∂P ⊥ p−∂P ⊥ qk 2 ≤ (1 − ) F (t) dt 1 (·) H H V H H L (0,ζ) [0, ζ ] 0 L2(0, ζ) Z 1 Z 1 p ζ(1 − ) + ζ (1 − ) F (t) dt + f(ζ) ≤ √ kF kL2(0, ζ) +ζ f(ζ) + (1 − ) F . 0 2 0 Then we conclude

(42) kPH⊥ p − PH⊥ qkV ≤ C ζ {kF kL2(−1, 1) + |f(ζ)|} Where C > 0 is an adequate constant.

3.6. Global Estimate of the Perturbation. For the global estimate of the dif- 2 ference recall kyk2 ≤ kyk1 for each y ∈ R and combine the estimates (39), (42) to get (43) 1 2 2 2 k p−qkV = {kPH (p−q)kV +kPH⊥ (p−q)kV } ≤ k PH p−PH qkV +k PH⊥ p−PH⊥ qkV √ ≤ 2 |f(0) − f(ζ)| + C ζ {kF kL 2(−1, 1) + |f(ζ)|}. In order to have continuous dependence of the solutions with respect to perturbations of the interface, it is direct to see that the finiteness of kF kL 2(−1, 1) needs to be required, and that conditions on the forcing term f behavior need to be stated. In the last line of inequality (43) the third summand needs f to be bounded in a neighborhood [0, δ) while the first summand demands it to be right continuous in 1 a neighborhood [0, δ) for some δ > 0. Recalling Hdiv(0, δ) = H (δ) in one dimension, the hypothesis that f = ∂Φ for some Φ ∈ Hdiv(0, δ) assumed in section 2.3 is sufficient to satisfy these conditions, however, it is not necessary. Finally, a repetition of the the same procedure for perturbations to the left, i.e. when ζ < 0 yields √

(44) k p − qkV ≤ 2 |f(0) − f(ζ)| + C |ζ|{kF kL 2(−1, 1) + |f(0)|} The section summarizes in the following result Theorem 3.3. Let F ∈ L2(−1, 1) and f ∈ C(−δ, δ) for some δ > 0, then

ζ (45) kp − q kV −−−→ 0. ζ→0 i.e. the sequence of perturbed solutions {qζ } converge strongly to the original one.

Proof. The estimate (44) together with the hypotheses on the forcing terms yield the desired results. Remark 4. It is important to stress that the successful analysis in the one dimensional case, heavily relies on the dimension itself. The characterization of the right space H and its orthogonal projections can not be done in a multiple dimensional setting, even in a very simple geometric domain such as the unit ball or the unit square. Also, the solutions provided by equations (40), (41) are possible only due to one dimensional framework. INTERFACE PERTURBATIONS AND NON-LINEARITY 13

Figure 3. Flattening of the Interface

Ωζ −ζ Ω 2 1 2

ζ Λ 1 Γ 2

Γ ζ +1 Λ 1 1

ζ Ω Ω 1 1

Remark 5. The estimates (43) and (44) heavily depend on the pointwise behavior of f, e.g. if f(x) ≡ C xs for s > 0 very small, the rate of convergence is very slow. It also reveals the nonlinear dependence of the solution q with respect to ζ.

4. A Simple Geometry in Multiple Dimensional Setting In this section we choose the simplest possible geometry in multiple dimensions in order to illustrate the nonlinearities that the phenomenon of interface geometric N−1 perturbation involves. Let Γ ⊆ R be open connected and Ω1 = Γ × (0, 1) and Ω2 = Γ × (−1, 0) i.e. the domain on the right of ﬁgure3. Also assume that the perturbation ζ is piecewise C1(Γ). We exploit the geometry deﬁning the fractional ζ bijective maps Λi :Ωi → Ωi for i = 1, 2 as follows

def XN − ζ(Xe ) (46a) Λi(Xe ,XN ) = (Xe , ) 1 − (−1)iζ(Xe ) Also deﬁne def X 1 (46b) Λ(X) = Λi(X) Ωi (X) i=1,2 Now set the new variables def (47) z = Λ(Xe ,XN ) · ebN , def (48) xe = Λ(Xe ,XN ) − Λ(Xe ,XN ) · ebN ebN .

0 4.1. Changes on Gradient Structure. For the maps Λi denote Λi its derivative or Jacobian matrix, then for i = 1, 2 holds " I 0 # 0 (49a) Λi = T . (1 − (−1)iz)∇e ζ 1 − (−1)iζ 14 FERNANDO A MORALES

Since kζkC(Γ) < 1 for functions belonging to T (Γ, Ω) (deﬁned in equation (1)), the absolute value of the determinant of the Jacobian matrix is given by

0 i i (49b) |det (Λi)| = 1 − (−1) ζ(xe ) = 1 − (−1) ζ(xe ). ζ For a scalar function we observe that whenever X ∈ Ωi the gradient has the following structure  z − (−1)i    i  ∇   ∇eX  I −(−1) ∇e ζ  ex     1 − (−1)iζ    (50a) ∂ =   ,  1  ∂   0T   ∂XN 1 − (−1)iζ ∂z in matrix notation

ζ ζ (50b) ∇X = Ai ∇x for all X ∈ Ωi and i = 1, 2.

4.2. Fractional Mapping and the H1(Ω) Space. From now on we endow the set T (Ω, Γ) with the norm W 1,∞(Γ) i.e. the sum of the essential suprema for the function and its gradient. Define the following change of variable Definition 4.1. For each element r ∈ H1(Ω) we define the Fractional Mapping operator by

def X −1 1 (51) T r = r ◦ Λi Ωi i=1,2 Lemma 4.2. Let r ∈ H1(Ω) then for each 1 ≤ ` ≤ N holds ∂ X ∂ (52) T r = r ◦ Λ−1 1 ∂ x ∂ x i Ωi ` i=1,2 ` i.e. the weak derivative does not have pulses/jumps on lower dimensional manifolds.

∞ Proof. Let ϕ ∈ C0 (Ω) then Z Z ∂ ∂ ϕ X −1 ∂ ϕ (53) h T r, ϕi 0 = − T r = − r ◦ Λ ∂ x D (Ω),D(Ω) ∂ x i ∂ x ` Ω ` i=1,2 Ωi ` X Z ∂ X Z = r ◦ Λ−1 ϕ − r ◦ Λ−1 ϕ (ν · e ) dS. ∂ x i 1 bi b` i=1,2 Ωi ` i=1,2 ∂Ωi We focus on the last two summands. Since ϕ = 0 on ∂Ω this implies Z X −1 − r ◦ Λ1 ϕ (νbi · be`) dS i=1,2 ∂Ωi Z −1 −1 = − { r ◦ Λ1 ϕ (νb1 · be`) + r ◦ Λ2 ϕ (νb2 · be`)}dS = 0. Γ −1 −1 The last equality holds since νb1 = −νb2 and Λ1 = Λ2 on Γ. Combining this fact with (53) we conclude (52).

Theorem 4.3. The map T is an isomorphism from H1(Ω) onto itself. INTERFACE PERTURBATIONS AND NON-LINEARITY 15

Proof. Since the application Λ is a bijection from Ω into itself the map T is clearly bijective and linear. For the calculation of the norms we use the Change of Variables theorem Z Z Z −1 2 2 0 2 |r ◦ Λi | = |r| |det (Λi)| ≤ 2 |r| i = 1, 2, ζ ζ Ωi Ωi Ωi where the last inequality holds due to (49b). Equivalently kT rk2 ≤ 2 krk2 0,Ωi ζ 0,Ωi i.e. T is a bounded operator in L2(Ω). For the derivative ﬁrst consider u ∈ C 1(Ω) −1 ζ −1 take 1 ≤ ` ≤ N − 1. For the vector function Λ i :Ω i → Ω i denote Λi,k its k-th component function, thus

2 Z ∂ 2 Z N ∂u ∂Λ−1 −1 X i,k 0 (u ◦ Λi ) = |det (Λi)| ∂ x` ζ ∂ xk ∂ x` Ωi Ωi k = 1 Z 2 ∂u ∂u i ∂ζ 0 = + 1 + (−1) z |det (Λi)| ζ ∂ x` ∂ z ∂ x` Ωi Z 2 Z 2 2 Z ∂u ∂u ∂ζ 2 2 ≤ 2 + 4 ≤ max {2, 4 kζkW 1,∞(Γ)} |∇u| . ζ ∂ x` ζ ∂ z ∂ x` ζ Ωi Ωi Ωi For the derivative with respect to z we get

def q 2 (54) C ζ = max {2, 4 kζkW 1,∞(Γ)}.

Then it follows

2 2 2 1 k∇(T u)k ≤ C k∇uk ζ ∀ u ∈ C (Ω), 0,Ωi ζ 0,Ωi for i = 1, 2, therefore

1 k∇(T u)k0,Ω ≤ C ζ k∇uk0,Ω ∀ u ∈ C (Ω). The inequality above extends to the whole space H1(Ω) by density of C 1(Ω) in H1(Ω). Finally, combining the ﬁrst and second parts we have

1 (55) k T r k1,Ω ≤ Cζ k r k1,Ω ∀ r ∈ H (Ω).

1 i.e. T is a bounded operator on H (Ω).

Corollary 1. The map T is an isomorphism from V onto itself.

−1 Proof. Observe that Λ1 |∂Ω1−Γ = I |∂Ω1−Γ, then T r = 0 on ∂Ω1 − Γ i.e. T is a bijection from V into itself and due to previous theorem the result follows. 16 FERNANDO A MORALES

4.3. The Fractional Mapping Operator on T (Ω, Γ). Consider the application T : T (Γ, Ω) → L(H1(Ω)), where ζ 7→ T (ζ) is deﬁned by equation (51). Since T (Γ, Ω) is not a linear space, only a convex set, T can not be linear; however T does not respect convex combinations either. Therefore, the nonlinearity of T does not lie only on its domain of deﬁnition but also on its algebraic structure. Clearly T (0) = I, we will show T is continuous at 0 in the pointwise topology.

1,∞ Lemma 4.4. Let {ζ n} ⊆ T (Γ, Ω) bounded in W (Γ) and such that kζ nkC(Γ) → 0, then 1 kT (ζ n)u − ukH1(Ω) → 0 , ∀ u ∈ C (Ω). Proof. First notice that for i = 1, 2 −1 i Λi (ζ n)(Xe ,XN ) = (xe, z(1 − (−1) )ζn(xe) + ζn(xe)).

Then the uniform convergence kζ nkC(Γ) → 0 implies −1 1 1 −1 1 (56) kΛ (ζ n) Ω + I Γ + Λ (ζ n) Ω − IkC(Ω) −−−−→ 0. 1 1 2 2 n→∞

(58) kT (ζ n)u − uk0,Ω → 0. For the H1(Ω)-convergence ﬁrst notice that due to the inequality (55) and deﬁnition (54) it follows

q 2 (59) kT (ζ n)uk1,Ω ≤ sup max{2, 4kζ nkW 1,∞(Ω)} kuk1,Ω. n ∈ N 1,∞ Where the supremum is finite because of the boundedness of {ζ n} in W (Γ). 1 Then the sequence {T (ζ n)u} has a weakly convergent subsequence in H (Ω) and due to (58) the weak limit must be u. Moreover, the Rellich-Kondrasov compactness theorem implies that the whole sequence converges weakly to the same limit u. Next we prove that the H1(Ω)-norms converge. The strong convergence in L2(Ω) is given by the Rellich-Kondrachov theorem, therefore we focus only on the derivatives. For any 1 ≤ ` ≤ N − 1 we have Z 2 Z ∂ −1 i 2 0 u ◦ Λi (ζ n) = |∂ù + ∂zu(1 + (−1) z)∂`ζ n| | det Λi(ζ n)| ∂ x` ζn Ωi Ωi Z i 2 0 = |∂ù + ∂zu(1 + (−1) z)∂`ζ n| | det Λi(ζ n)| ζn Ωi Z i 2 i = |∂ù + ∂zu(1 + (−1) z)∂`ζ n| (1 − (−1) ζ n). ζn Ωi INTERFACE PERTURBATIONS AND NON-LINEARITY 17

∂ u 2 1 On one hand, it is clear that the integrand converges to | | Ωi , on the other ∂ x` hand, we have the estimate

i 2 i ∂ù + ∂zu(1 + (−1) z)∂`ζ n| (1 − (−1) ζ n)1 ζn Ωi 2 i 2 2 2 2 2 ≤ 2 {|∂ù| + |1 + (−1) z||∂`ζ n| |∂zu| } 2 ≤ 4 {|∂` u| + 2 kζ nkW 1,∞(Γ)|∂z u| } 2 2 1 ≤ 4 max{1 + 2 sup kζ nkW 1,∞(Γ)} |∇u | ∈ L (Ω) n ∈ N Thus, Lebesgue’s dominated convergence theorem yields 2 2 ∂ ∂u (60) (T (ζ n)u) −−−−→ 1 ≤ ` ≤ N − 1, for i = 1, 2. ∂ x n → ∞ ∂ x ` 0,Ωi ` 0,Ωi For the derivative with respect to z we get Z 2 Z 2 ∂ −1 ∂u i 0 u ◦ Λi (ζ n) = 1 + (−1) ζ n |det (Λi(ζ n))| ∂ z ζn ∂z Ωi Ωi Z 2 ∂u i 3 1 = 1 + (−1) ζ n Ωζn Ω ∂z i ∂u ∂u Again, the integrand converges to | |2 1 and it is bounded by 2 | | 2 which is ∂z Ωi ∂z an element of L1(Ω). Hence, the Lebesgue dominated convergence theorem yields 2 2 ∂ ∂ u (61) (T (ζ n)u) −−−−→ , for i = 1, 2. ∂z n → ∞ ∂z 0, Ωi 0, Ωi The equations (60) and (61) give the convergence of the L2(Ω)-norms of the gradients k∇T (ζ n)uk 0,Ω → k∇uk 0,Ω and then kT (ζ n)uk 1,Ω → kuk 1,Ω. This fact 1 together with the weak convergence in H (Ω) imply kT (ζ n)u − uk 1,Ω → 0. 1,∞ Theorem 4.5. Let {ζ n} ⊆ T (Γ, Ω) bounded in W (Γ) and such that kζ nkC(Γ) → 0, then 1 kT (ζ n) r − rkH1(Ω) → 0 , ∀ r ∈ H (Ω) i.e. T (ζ n) converges to the identity I in the strong operator topology. 1 1 Proof. We use the standard density argument. Let r ∈ H (Ω), take {uj } ⊆ C (Ω) such that kuj − rk1,Ω → 0. Recall definition (54) and inequality (55), then

k T (ζ n) r−rkH1(Ω) ≤ k T (ζ n) r−T (ζ n)uj kH1(Ω)+k T (ζ n)uj −uj kH1(Ω)+k uj −rkH1(Ω)

≤ (1 + k T (ζ n)k)kuj − rkH1(Ω) + k T (ζ n)uj − uj kH1(Ω) q ≤ (1 + sup max {2, 4 kζ kkW 1,∞(Γ)}) k uj − rkH1(Ω) + k T (ζ n)uj − uj kH1(Ω) k ∈ N N Fix j ∈ such that the ﬁrst summand of the right hand side is less than 2 . Due to lemma 4.4 there exists N ∈ N such that n ≥ N implies the second summand on the right hand side of the expression above is less than 2 and the proof is complete. 1,∞ Corollary 2. Let {ζ n} ⊆ T (Γ, Ω) bounded in W (Γ) and such that kζ nkC(Γ) → 0, then −1 T (ζ n) r − r 1 −−−−→ 0 H (Ω) n→∞ i.e. the sequence of inverse operators converge. 18 FERNANDO A MORALES

Proof. Repeating the techniques exposed in Lemma 4.4 and in theorem 4.5 applied to the fractional bijective maps {Λi(ζ n)} deﬁned in (46a) the result follows.

Remark 6. Using theorem 4.5 it can be proved that k T (ζ n) −  kL(C 1(Ω),L2(Ω)) → 0 1,∞ 1 2 as kζnkC(Γ) → 0 if {ζn} is bounded in W (Γ). Here  : C (Ω) ,→ L (Ω) is the def 1 embedding operator (ϕ) = ϕ and T (ζ n) is regarded as an operator from C (Ω) to L2(Ω). Also, using the same technique in obtaining the estimate (57) we can 2 1 show k T (ζ n) −  kL(C2(Ω),H1(Ω)) → 0 with  : C (Ω) ,→ H (Ω); i.e. in order to get convergence in the norm of the operators higher degrees of regularity are needed. 4.4. The Flattened Problem. Consider the problem (8b) subject to the changes of variable described above. Introducing (49b) and (50) in each summand of left hand side in (8b) we have Z Z Z 0 ζ T ζ i ∇X q·∇X r = ∇X T q·∇X T r |det Λi| = (Ai ) Ai ∇xT q·∇xT r (1−(−1) ζ). ζ Ωi Ωi Ωi With   (1 − (−1)iζ)I (−1)i(z − (−1)i)∇e ζ (62) (Aζ )T Aζ =   . i i  T i 2   i i |(z − (−1) )∇ζ| + 1 (−1) (z − (−1) )∇e ζ 1 − (−1)iζ Due to corollary1 the quantiﬁers ∀ T r ∈ V and ∀ r ∈ V are equivalent, therefore we conclude that the solution q of problem (8b) satisﬁes the variational statement

X Z 1 − (−1)i ζ (63) q ∈ V : (Aζ )T Aζ ∇T q · ∇r i−1 i i i = 1, 2 Ωi Z 1 X Z = (T f) r d S + (1 − (−1)i ζ )(TF ) r , ∀ r ∈ V. Γ |(−∇e ζ, 1)| i = 1, 2 Ωi

−1 In the ﬁrst summand of the left hand side, the notation T f stands for f ◦ Λ1 −1 −1 −1 or f ◦ Λ2 indistinctively since Λ1 = Λ2 on Γ. The surface integral summand implicitly uses the fact that the upwards unitary vector normal to the surface Γ ζ (−∇e ζ, 1) def is given by nb = . Declaring % = T q the problem (63) is equivalent to |(−∇e ζ, 1)|

X Z 1 − (−1)i ζ (64) % ∈ V : (Aζ )T Aζ ∇% · ∇r i−1 i i i = 1, 2 Ωi Z 1 X Z = (T f) r d S + (1 − (−1)i ζ )(TF ) r , ∀ r ∈ V. Γ |(−∇e ζ, 1)| i = 1, 2 Ωi Next, we focus on some properties of the involved matrices.

ζ Lemma 4.6. For i = 1, 2 the inverse matrix of Ai is given by " i # ζ −1 I (1 − (−1) z)∇e ζ (65) (Ai ) = 0T 1 − (−1)iζ

Proof. By direct calculation INTERFACE PERTURBATIONS AND NON-LINEARITY 19

ζ Corollary 3. Let (xe, z) ∈ Ωi be ﬁxed with i ∈ {1, 2}. Then the linear operator ζ −1 RN ξ 7→ (Ai (x)) ξ from into itself endowed with the canonical inner product satisﬁes

ζ −1 q 2 ζ N (66) k(Ai (x)) kL(R ) ≤ N + 3 + 4 kζkW 1,∞(Γ) , for all x ∈ Ωi , and i ∈ {1, 2} Proof. We compute the Frobenius norm of the operator adding the squared inner product norms | · | of each column vector in (65). This gives ζ −1 2 i 2 2 i 2 k(A (x)) k N ≤ (N − 1) + (1 − (−1) z) |∇ζ(x) | + (1 − (−1) ζ) . i L(R ) e Since z, ζ(xe) ∈ [−1, 1] for all x = (xe, z) ∈ Ω we estimate |1 ± z| ≤ 2 and |1 ± ζ| ≤ 2. 1,∞ The gradient of ζ is estimated by the W (Γ)-norm and the result follows. i ζ T ζ RN Proposition 1. The matrices (1 − (−1) ζ)(Ai ) Ai are uniformly coercive in for i = 1, 2 i.e. there exists e(ζ) > 0 such that 2 i ζ T ζ (67) e(ζ) |ξ| ≤ (1 − (−1) ζ)(Ai ) Ai ξ · ξ RN For all ξ ∈ and each (xe, z) ∈ Ω. N Proof. Let x ∈ Ωi be ﬁxed and ξ ∈ R unitary then 2 (1 − (−1)iζ)(Aζ )T Aζ ξ · ξ = (1 − (−1)iζ)Aζ ξ · Aζ ξ ≥ (1 − kζk ) Aζ ξ i i i i C(Γ) i 2 ζ 1 − kζkC(Γ) 1 − kζkC(Γ) ≥ (1 − kζkC(Γ)) min A η = ≥ . i ζ −1 2 2 |η|=1 k(A ) k N + 3 + 4 kζk 1,∞ i L(RN ) W (Γ)

def 1−kζkC(Γ) Where, the last inequality comes from corollary3. Defining e(ζ) = N+3+4 kζk2 W 1,∞(Γ) the statement (67) holds. Corollary 4. The form [ ·, ·]: V × V → R Z i def X 1 − (−1) ζ (68) [κ, π] = (Aζ )T Aζ ∇κ · ∇π i−1 i i i = 1, 2 Ωi defines an inner product on V which induces the same topology as the induced by the standard inner product on H1(Ω). Proof. The form (68) is well-defined since the application i ζ T ζ (xe, z) ∈ Ωi 7→ (1 − (−1) ζ(xe))(Ai (xe, z)) Ai (xe, z) ∞ N×N for i = 1, 2 is in L (Ω i, R ). Clearly the form is bilinear and symmetric. For the 2 ζ 2 2 ζ 2 continuity we have [κ, π] ≤ kAi k∞ k∇κk0, Ω k∇πk0, Ω ≤ kAi k∞ kκ k1, Ω kπk1, Ω 2 ζ 2 2 for all κ, π ∈ V ; in particular [κ, κ] ≤ kAi k∞ kκ k1, Ω. The homogeneous condition of the induced norm comes from the uniform coercivity of the matrices shown in proposition1. Hence

e(ζ) 2 2 kκk1,Ω ≤ e(ζ) k∇κk0, Ω ≤ [κ, κ] ∀ κ ∈ V. 1 + CΩ

Where CΩ is the Poincar´econstant associated to the domain Ω valid for all elements of V given the boundary conditions. Therefore, the induced norms are equivalent. Theorem 4.7. The problem (64) is well-posed. 20 FERNANDO A MORALES

Proof. The equivalence of norm induced by the inner product [ ·, ·] to the standard H1(Ω)-norm on V is shown in corollary4, therefore the well-posedness of problem (64) follows from Lax-Milgram’s lemma.

4.5. Geometric Perturbation and Inner Products. This section is aimed to analyze the highly nonlinear impact of the geometry in terms of the inner product. Consider the bounds

ζ −1 −1 ζ (69) kp − q kV ≤ kp − T (ζ)p kV + kT (ζ)p − q kV

The ﬁrst summand converges due to theorem 4.5. For the convergence of the second summand, due to corollary2 it is equivalent to prove

ζ ζ (70) kp − T (ζ)q kV = kp − % kV → 0 as kζkC(Γ) → 0 with kζkW 1,∞(Γ) bounded.

We are to estimate the norm above by comparing the problems (8a) and (64). In problem (64) the eﬀect of the geometry on the inner product structure is fully contained in the matrices

i ζ T ζ (71) {(1 − (−1) ζ)(Ai ) Ai : x ∈ Ωi} i = 1, 2. Notice this is a family of symmetric and therefore diagonalizable matrices. However, they depend on the point x ∈ Ω and, in general, they do not commute for x, x0 ∈ Ω diﬀerent. Therefore, it can not be assured that the family (71) is simultaneously diagonalizable. Observe that the matrices (50a) have entries multiplied by the 1 factors i , for i = 1, 2 respectively. This implies that the maps induced 1 − (−1) ζ(xe) by the matrices are not linear with respect to the perturbation ζ. The following hypothetic assumptions illustrate the nonlinearity of the dependence. (i) Suppose that ζ is a piecewise linear aﬃne function. Although ∇e ζ is piecewise i ζ T ζ constant, the map x 7→ (1 − (−1) ζ)(Ai ) Ai is not piecewise constant. (ii) Assume that the family of matrices (71) is diagonal. Testing the problems (8a) and (64) with p − % and subtracting them yields

  µ1 ... 0 X Z ∇p − ∇ % (72) ·{ ∇p −  . . .  ∇ %} i−1  . . .  i = 1, 2 Ωi 0 . . . µN Z X Z = F (p − %) − (1 − (−1)i ζ)(TF )(p − %) dS Ω i = 1, 2 Ωi Z Z 1 + f(p − %) dS − (T f)(p − %) dS Γ Γ |(−∇e ζ, 1)|

Where µ1, . . . , µN are the eigenvalues. Nevertheless µj = µj(x) for 1 ≤ j ≤ N i.e. they depend on the position x within the domain Ω. We close this section reviewing the simplest possible scenario.

Fractional Mapping of Domains in the One Dimensional Case. In this section we analyze the fractional mapping technique in the one dimensional setting. INTERFACE PERTURBATIONS AND NON-LINEARITY 21

For this case the problem (64) reduces to

1 Z 0 1 1 Z 1 (73) ∂% · ∂r + ∂% · ∂r = f(ζ) r(0) 1 + ζ −1 1 − ζ 0 (−1)i+1 X Z 2 x − ζ + (1 − (−1)iζ) r(x) F dx (−1)i−1 1 − (−1)iζ i=1,2 2 In order to get a-priori estimates the left hand side of the problem (73) suggests testing equations (27a) and (73) with the function (74) 1 1 1 1 (p − %) 1 + (p − % )1 = p − { % 1 + % 1 }. 1 + ζ (−1, 0] 1 − ζ [0, 1) 1 + ζ (−1, 0] 1 − ζ [0, 1) However, the test function presented above (as the second summand in the right hand side illustrates) is not eligible, because it does not belong to V unless ζ = 0 or %(0) = 0. The ﬁrst condition removes the perturbation leaving the original problem

(27a) and the second can not be assured. Any other attempt of estimating kp−%kV demands test functions equivalent to (74), because of the coefficients disagreement in problems (27a) and (73). Of course such piecewise functions are not in the test space V due to the trace continuity requirements. In the one dimensional case, the fractional mapping technique defines an inner product much easier to understand than the corresponding to the multidimensional case (64). It is clear that the dependence of the inner product with respect to the perturbation is piecewise linear fractional as the map Λ : (0, 1) → (0, 1) itself. Additionally, direct calculations can be done to find explicitly, the dependence of the eigenvalues associated to the problem (27b). This dependence also turns out to be piecewise fractional on ζ, closely related to Λ. For a deep discussion on boundary perturbation of the Laplace eigenvalues see [8,9]. Finally, the question of strong convergence (70) can be solved using the Rellich- Kondrachov theorem. However, this is not a constructive result and it does not yield explicit estimates such as inequality (44) in section 3.6.

5. Concluding Remarks and Discussion The present work yields several conclusions and open questions. (i) In section 2.3, extra hypothesis of regularity on the forcing terms involved were introduced in order to conclude weak convergence in a first step, and strong convergence in a second one. Also, for the geometric perturbations ζ ∈ T (Ω, Γ) it is not enough to have convergence in C(Γ), there is also need for boundedness in W 1,∞(Γ). (ii) The conditions of convergence for the interface are acceptable in the context of saturated porous media fluid flow. Moreover, for the modeling of saturated fluid flow through deformable porous media in the elastic regime, these conditions are natural, because for high gradients of deformation the elasto-plastic and plastic regimes start taking place, see [17]. (iii) Mapping the perturbed domains with fractional applications as in section 4, decomposes the nonlinearity of the question in two parts: the fractional mapping operator T : T (Ω, Γ) → L(H1(Ω)) which is clearly nonlinear, and the effect of the geometric perturbation on the inner product that the problem (8b) defines on V . The latter is reflected in the matrices (62) of the problem (64) above. 22 FERNANDO A MORALES

(iv) Although the operators T (ζ) converge in the strong operator topology as the perturbations ζ ∈ T (Ω, Γ) converge, according to the hypothesis of theorem 4.5, it is an abstract statement. There are no explicit estimates depending on kζkW 1,∞(Γ), analogous to inequality (44) presented in section 3.6. (v) The inner product that a geometric perturbation implicitly defines on the function space V is the most important nonlinearity of the problem. It introduces a notion of orthogonality in the space which is hardly comparable with the standard one, beyond the topological equivalence of the induced norms. (vi) The fractional mapping technique is a much simpler approach than the local charts strategy used in trace theorems. Its main contribution in this work, consists in exposing the challenges of the convergence rate question as well as the non-linearities involved, in a much neater way than the local charts approach. (vii) The strong convergence statements attained in section 2.4, realizing problem (8a) as the strong limit of the family of problems (8b), suggest numerical experimentation and a-posteriori estimates as the most feasible approach to gain insight in the convergence rate question as well as dependence with respect to the norms W 1,∞(Γ) and C(Γ). (viii) The solution of the one dimensional case presented in section3, using orthogonal decompositions on carefully chosen subspaces, illustrates the complexity of the convergence rate problem. This fact becomes even more dramatic due to the strong dependence on the one dimensional setting. (ix) In both settings, multiple and one dimensional, the necessity of testing the variational statements with functions of the structure (74) (i.e. discontinuous across the interfaces), to obtain explicit a-priori estimates of the difference p − q is self- evident. Such functions break the linear structure of the domain V and obey to the disagreement of scaling coefficients in problems (8a), (8b). 2 1 2 (x) Both of the mixed variational formulations: L -H and H div-L demand coupling conditions on the function spaces for the trace on the interfaces Γ, Γ ζ . These conditions do not allow testing with discontinuous functions such as 74. However, 2 1 2 a mixed formulation setting L -H in one region, namely Ω1 and H div-L in the other region, as the one introduced in [16], does not require continuity of the test functions across the interfaces. Hence, this is the formulation where convergence rate estimates (implicit or explicit, given the nonlinearity of the problem) can most likely be attained.

6. Acknowledgments The author thanks to Universidad Nacional de Colombia, Sede Medell´ınfor partially supporting this work under project HERMES 17194 and the Department of Energy USA for partially supporting this work by grant 98089.

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